Direct migration

The correlated wavefield $\tilde{R}$ is not usable by the majority of available reflection migration data tools. The source axis summation explained above does not remove all of the potential time delays. However, field data can still be migrated with a scheme that includes separate extrapolation and correlation (for imaging) steps. Artman and Shragge (2003) shows the applicability of direct migration for transmission wavefields with a shot-profile algorithm. Artman et al. (2004) provides the mathematical justification (for zero phase source functions). Simply stated, both Fourier domain extrapolation across a depth interval and correlation are diagonal square matrices, and thus commutable. This means that the correlation required to calculate the earth's reflection response from transmission wavefields can be performed after extrapolation with the shot-profile imaging condition Rickett and Sava (2002)  

$$i_z({\bf x},{\bf h})= \delta_{{\bf x},{\bf x}_r} \; \sum... ...\omega) D_z^*({\bf x}_r-{\bf h}; {\bf x}_{s_k},\omega) \; ,$$ (9)

where T is used for both upcoming, U, and downgoing, D, wavefields.

Figure  pictorially demonstrates how direct migration of field passive seismic data fits into the framework of shot-profile migration to produce the 0^th and 1^st depth levels of the zero offset image. The sum over frequency has been omitted to reduce complexity in the figure. Also, after the first extrapolation step, with the two different phase-shift operators, the two transmission wavefields are no longer identical, and can be redefined U and D. This is noted with superscripts on the T wavefields at depth.

  

Figure 5: Equivalence of shot-profile migration of reflection data and direct migration of passive wavefields. $T(\xi,t)$ are the wavefields of equation 1. ${\bf x}_s$ has a similar meaning to $\xi$. Only first and second levels of the iterative process are depicted. $\sum_{{\bf x}_s}$ and $\sum_\omega$ produces the image $i_z({\bf x}_r,{\bf h})$ for both methods.

Shot-profile migration becomes planewave migration if conventional shot-gathers are summed for wavefield U, and a horizontal plane source is modeled for wavefield D. Partial summation of conventional shot-records will introduce cross-talk into the image. Only by summing enough shots so that their destructive interference cancels out their cross-talk can one produce a high quality image. For raw passive data, the sum over sources leads to an areal wave with complicated temporal topography. Moving the sum over shots in the imaging condition of equation 9 to operate on the wavefields rather than their correlation, changes shot-profile migration to something akin to planewave migration which I will call wavefront migration. Like planewave migration, after even a few wavefields have been combined, the best course of action is to sum all the sources to attain good areal coverage of the source wavefront to minimize cross-talk. Figure  shows the change source summation has on both conventional shot migration and direct passive migration. Notice the parameterization of $T(\tau)$ meaning field data (where the depth subscript displaces the use of T_f). Importantly, the data input into the migration needs to have the late lags windowed before input into the migration as they have no correspondence to the subsurface structure. This can be accomplished by any of the three options discussed above: correlation followed by windowing, DFT followed by subsampling, or stack followed by DFT and correlation.

  

Figure 6: Equivalence of direct migration with simultaneous migration all shots in a reflection survey. $T(\tau)$ is the field data wavefield of equation 3. Only first and second levels of the iterative process are depicted. $\sum_\omega$ produces the image $i_z({\bf x}_r,{\bf h})$ for both methods.